theorem Th20:
  F is with_the_same_dom & E c= dom(F.0) & (for m be Nat holds F.m
  is E-measurable & G.m= (F.m)|E) implies G.n is E-measurable
proof
  assume that
A1: F is with_the_same_dom and
A2: E c= dom(F.0) and
A3: for m be Nat holds F.m is E-measurable & G.m= (F.m)|E;
  dom(F.n) = dom(F.0) by A1;
  then dom(F.n) /\ E = E by A2,XBOOLE_1:28;
  then (F.n)|E is E-measurable by A3,MESFUNC5:42;
  hence thesis by A3;
end;
